An adsorption-permeability model of coal with slippage effect under stress and temperature coupling condition

Journal of Natural Gas Science and Engineering 71 (2019) 102983

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An adsorption-permeability model of coal with slippage effect under stress and temperature coupling condition

T

Bobo Lia,b,c,∗, Kang Yangd, Chonghong Rena, Jianhua Lia, Jiang Xud a

College of Mining, Guizhou University, Guiyang, 550025, China The National Joint Engineering Laboratory for the Utilization of Dominant Mineral Resources in Karst Mountain Area, Guizhou University, Guiyang, 550025, China c State Key Laboratory of Mining Disaster Prevention and Control co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, 266590, China d State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing, 400044, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Stress and temperature coupling Permeability Adsorption Slippage effect Excess adsorption

Temperature and slippage effect, two important factors affecting coalbed methane (CBM) production, both have an intuitive effect on gas adsorption in coal seams and coal deformation and gas seepage processes. In order to simulate the process of CBM exploitation, isothermal adsorption tests at different temperatures and seepage tests under rising pore pressure were carried out. In this study, the modified Langmuir model considering the effects of temperature and excess adsorption was established. On this basis, the amount of adsorption deformation was calculated. The results show that the amount grows with the rise of pore pressure and is negatively related to temperature. In addition, a temperature mutation coefficient (γT) was introduced to characterize the response of coal permeability to temperature, and thermal expansion, thermal cracking, adsorption deformation induced by temperature and slippage effect were combined to establish a coal permeability model under the coupling of stress and temperature. Meanwhile, a helium parallel seepage test was carried out under the same conditions to study the effects of slippage and adsorption expansion. It is found that the rise of pore pressure will reduce permeability under the action of adsorption or slippage effects, and the rise of temperature will enhance the permeability. Finally, according to the experimental data, the adsorption-permeability model achieves better fitting results, compared with other models. The model can provide certain theoretical support for CBM exploitation.

1. Introduction Coalbed methane (CBM), a form of natural gas extracted from coal beds, has a great development and utilization value. Despite the abundant CBM resources in China, the permeability of highly gassy coal seams is generally low, and most of CBM is stored deep in the ground, which increases the difficulty of CBM exploitation (Luo et al., 2011). With the gradual depletion of shallow resources, the exploitation of deep resources has gradually become a new normal (Xie et al., 2015), and the coal seam temperature increases linearly with the gradual increase of CBM mining depth. Meanwhile, the underground geotherm sometimes exceeds the normal value 25 °C/km. The temperature will affect deformation and permeability characteristics of coal (Guan et al., 2018; Perera et al., 2011; Zhu et al., 2011b). The adsorption characteristics of coal affect the migration law of CBM and are of great significance for estimating CBM content (Harpalani et al., 2010). The slippage effect plays an important role in the process of CBM ∗

exploitation, but it is often neglected (Li et al., 2014). Therefore, the study on the migration law of CBM under stress and temperature coupling condition is of great significance for CBM exploitation by gas injection. Coal is a dual-porosity medium consisting of pores and fractures (Meng et al., 2018). Gas desorption, diffusion and seepage in coal seams are dynamic and complex processes that accompany CBM exploitation. During CBM extraction, free gas in coal fractures is discharged due to the pressure gradient, and adsorbed gas in the coal matrix is gradually desorbed with the decrease of pressure. Driven by the internal and external concentration difference of the matrix, the free gas is diffused to coal fractures and discharged. In this process, the coal matrix is contracted and the effective seepage channel is increased due to gas adsorption and desorption of coal, thus increasing the permeability (Wang et al., 2018). Therefore, the adsorption characteristics of coal are an important factor affecting the deformation of coal and the law of gas migration. In fact, the adsorption characteristics are affected by many

Corresponding author. College of Mining, Guizhou University, Guiyang, 550025, China. E-mail address: [email protected] (B. Li).

https://doi.org/10.1016/j.jngse.2019.102983 Received 24 April 2019; Received in revised form 14 August 2019; Accepted 30 August 2019 Available online 03 September 2019 1875-5100/ © 2019 Elsevier B.V. All rights reserved.

Journal of Natural Gas Science and Engineering 71 (2019) 102983

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In order to quantitatively study the evolution law of coal permeability during CBM exploitation, a series of permeability prediction models with different boundary conditions have been established (Lu et al., 2016). Under uniaxial strain conditions and matchstick geometry, Gray et al. (1987) was the first to incorporate factors influencing matrix shrinkage and effective stress into the permeability model, after which many theoretical permeability models were presented by the same assumptions (Palmer and Mansoori, 1998; Shi and Durucan, 2004, 2014; Cui and Bustin, 2005; Pan and Connell, 2011). Palmer and Mansoori (1998) considered the factors influencing matrix shrinkage and effective stress and established the permeability prediction model (P&M model) from the perspective of strain. Based on the P&M model, Shi and Durucan (2004) derived the permeability model (S&D model) by introducing the compressibility coefficient model. Cui and Bustin (2005) established a model (C&B model) by quantitatively analyzing effects of gas adsorption and volumetric strain on coal seam permeability. Pan and Connell (2011) applied an anisotropic expansion model to the S&D model to describe the permeability behavior. These are all common permeability models, most of which assumed overburden loads to be constant. Several scholars have proposed permeability models with the triaxial stress assumption (Robertson and Christiansen, 2008; Connell et al., 2010; Guo et al., 2014). The triaxial stress model can be transformed to a uniaxial strain model by replacing specific boundary conditions (Liu et al., 2011). At present, most relevant researches on coal permeability models assume coal to be under uniaxial strain and constant axial pressure. Few permeability models explore the condition of constant effective stress, and most adsorption expansion deformation models adopt the simplified Langmuir model which does not consider effects of excess adsorption and temperature. Hence, the prediction accuracy of the model cannot be guaranteed. As mentioned above, previous studies on adsorption and permeability characteristics of coal under the coupling of stress and temperature are mostly focused on the experimental qualitative research. Generally, a single factor of either coal adsorption or permeability characteristics was studied under temperature. However, there are relatively few studies on the coupling effect of temperature and excess adsorption on coal adsorption and permeability characteristics, and the theoretical analysis of the influence of temperature on the slippage effect is rarely reported, either. The authors have conducted related research earlier and established a slippage model considering the temperature effect (Li et al., 2019), but the coupling effect of temperature and excess adsorption on the adsorption and seepage characteristics of coal are not considered in the model. In this study, to more accurately describe the effects of adsorption, temperature and pore pressure on deformation and permeability, an isothermal adsorption experiment and a seepage experiment were carried out under different temperature conditions. Based on the isothermal adsorption experiment, a Langmuir model considering the correction of temperature and excess adsorption was established. This study modified the amount of coal deformation under the temperature effect according to the adsorption model, and established the model of slippage effect and adsorption swelling considering the temperature effect. An adsorptionpermeability model considering slippage effect and adsorption swelling under stress and temperature coupling condition was established and verified by the experimental results, which provides a theoretical basis for deep CBM exploitation.

factors, including temperature, gas pressure (Zhang, 2008; Pan et al., 2012), moisture (Guo et al., 2015), coal composition and metamorphism (Dutta et al., 2011; Ju et al., 2009). With the increase of CBM mining depth, temperature significantly affects the adsorption characteristics, which has been extensively investigated in numerous studies. The rise of temperature promotes the gas desorption which induces coal deformation at the macro level and alters the crack size of coal at the micro level. Temperature also has a negative effect on the adsorption and swelling of coal (Mukherjee and Misra, 2018). The experimental value of methane adsorption can be calculated by various models, among which the Langmuir equation is the most commonly used one. However, the equation does not reflect the coupling between temperature and pressure. Anderson et al. (2011) established a Langmuir model considering the effect of temperature, but it failed to consider the effect of excess adsorption. Since gas generally remains in the supercritical state in the process of deep CBM extraction, the adsorption amount calculated in the laboratory belongs to excess adsorption capacity which cannot reflect the true adsorption capacity of coal seam (Pini et al., 2010). At low pressure, the excess adsorption is similar to the absolute adsorption. Nevertheless, the error of excess adsorption cannot be ignored as the pressure rises (Weniger et al., 2010). Tang et al. (2016) proposed a Langmuir model considering the correction of excess adsorption capacity, and then they validated the model. Temperature has an inhibitory effect on coal swelling resulted from gas adsorption. The rise of temperature reduces the amount of adsorbed gas, thus lowering the adsorption-caused strain. In recent years, many researchers have established swelling isotherm models (Day et al., 2011; Liu and Harpalani, 2013), but few models consider the coupling between temperature and pressure. In the process of CBM exploitation, the effects of effective stress, adsorption swelling and slippage effect on coal permeability changes dynamically. At the initial stage of mining, the effective stress determined by pore pressure and overlying strata pressure plays a dominant role. With the lowering of pore pressure, the effective stress increases and coal permeability decreases. As pore pressure further decreases, the shrinkage and slippage effect of coal matrix dominate, and meanwhile coal permeability is enhanced. (Meng et al., 2018). Temperature is an important factor affecting CBM storage and migration. The process of temperature affecting coal deformation and permeability is complicated (Yin et al., 2013). Temperature can cause coal deformation through thermal expansion, coal damage and inhibition of coal adsorption (Otake and Suuberg, 1997; Zhu et al., 2011b; Qu et al., 2012). With the rise of temperature, gas possesses stronger molecular activity. It becomes harder for gas to adsorb on coal, so the shrinkage effect is enhanced (Sakurovs et al., 2008; Charoensuppanimit et al., 2015). The phenomenon of thermal swelling and thermal cracking will be aggravated with the rise of temperature, which could promote coal deformation and affect its internal structure (Akbarzadeh et al., 2014; Teng et al., 2016). However, coal cleats and fractures are the main channels for gas seepage, so temperature-induced coal deformation inevitably alters coal permeability. The slippage effect is an important factor affecting coal permeability that is determined by the gas relative free path and pore size, so variations in pore width and pore pressure affect slippage effects (Wang et al., 2014). Under various stress conditions, sorption-induced swelling and shrinkage will change cleat width and slippage effects even under fixed effective stress or confining pressure (Pan et al., 2009; Wang et al., 2014; Zhou et al., 2016). Although slippage increases coal permeability, it is often overlooked. Given the coupling effect of slippage and matrix swelling on permeability, researchers have carried out a series of experimental and theoretical studies (Wang et al., 2014; Zou et al., 2016). Temperature and pore pressure jointly impact the change in matrix pores of coal and gas relative free path, so temperature is an important factor influencing slippage effect (Wang et al., 2010). However, there are just relatively few theoretical researches on the influence of slippage on permeability under the coupling of stress and temperature.

2. Experimental 2.1. Sample preparation Taken from K2 coalbed of Songzao coal factory in Chongqing, China, the coal sample was prepared into 60–80 meshes pulverized coal particles in light of The Isothermal Adsorption Experimental Method (GB/ T1960-2004). The coal sample in seepage test was prepared in strict accordance with the method of coal mechanical property 2

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(1) Air tightness check: The initial pressure was set to 3 MPa by opening the valve and adding the gas. If the pressure value remained constant after 12 h, it indicated that the test device was well sealed and the next test operation could be carried out. (2) Free space measurement: After the vacuum pump was turned on for 4 h, the intake valve was opened to inject He gas into the reference cylinder. After stabilization, the gas storage tank and the adsorption tank were connected, and the free space volume of sample cylinder was calculated by the gas state equation. (3) Adsorption test: The residual gases were removed from the system through vacuuming again. Next, methane was injected into the gas storage cylinder until the gas pressure stabilized. Then, the pressure value was recorded. At this time, the switch valve between the gas storage tank and the adsorption tank was opened to allow coal to reach adsorption equilibrium. The adsorption test was repeated by changing the gas pressure. (4) The sample was replaced to perform the isothermal adsorption test at 40 °C, 50 °C and 60 °C according to the previous 3 steps.

determination. Because the raw coal in Songzao Coal Mine belongs to extremely soft weak rock, presence of natural fractures posed difficulties during the extraction of intact test samples. Even if the raw coal was cored, the samples would fail to be representative because they were just individual blocks in a single coal seam. It is known that coal permeability consists of matrix permeability and fracture permeability, of which the latter contributes a lot to the effective permeability of coal. Considering that coal seams are generally highly heterogeneous, it is impossible to obtain a primary coal seam with the same fracture system and permeability (Su et al., 2019). Instead, it will be beneficial to perform experiments on coal samples with the same or similar permeability and to analyze effects of other factors (such as pore pressure and temperature) on the slippage effect and adsorption expansion. According to previous studies, briquette shares consistency with raw coal in terms of gas permeability variation (Jasinge et al., 2011; Wu et al., 2018; Su et al., 2019; Chu et al., 2017). Therefore, the briquette was used in the experiment. It is noteworthy that although the briquette cannot present the original reservoir structure (Sander et al., 2016), parallel tests of CH4 and He with constant effective stress under different temperature conditions are applicable to studying effects of temperature and pore pressure on the slippage effect and adsorption expansion. First, after the raw coal was crushed, the coal particles with sizes of 60–80 meshes were selected. Next, 100 MPa of pressure was applied to the coal particles by a rigid testing machine for 20 min to make cylindrical standard 50 mm × 100 mm coal samples. Finally, the prepared samples were placed in an oven at 80 °C for 24 h before they were placed in a desiccator. In order to ensure the representativeness of test samples, the tested selected three samples whose proximate analysis, pore and fracture conditions are listed in Tables 1 and 2, respectively. It can be seen from Tables 1 and 2 that the basic component, pore and fracture parameters of the three samples are in good agreement with each other, indicating their representativeness.

2.3.2. Permeability measurement To simulate the process of gas production in deep coal seams and investigate the evolution law of adsorption and seepage under the coupling of stress and temperature, with CH4 and He being the experimental gases, experiments were carried out at different temperatures (30 °C, 40 °C, 50 °C and 60 °C) in the gas loading process. During the process, the effective stress was set at 5.7 MPa for eliminating the effect of stress. Finally, the experimental data were substituted into the model to calculate the permeability variation induced by slippage effect and adsorption swelling at different temperatures. Specific test steps are as follows (Li et al., 2017, 2019): (1) Installation of specimen and vacuum degassing: First, the surface of the dried raw coal sample was uniformly smeared with 704 silicone rubber. Next, it was put into in the triaxial cell where a hot pyrocondensation pipe shrank to touch the coal wall closely. After completing the above operation, the radial extensometer was installed in the middle of the whole specimen and was debugged to ensure complete and correct connection with the computer. Next, the axial retainer was installed, and the coal sample and upper and lower parts of the pressure head were fixed. Meanwhile, the triaxial pressure chamber shall be installed by first tightening all screws and then connecting the ventilation pipe with the gas cylinder. After test sample installation and air tightness check were completed, the sample was vacuum degassed by a vacuum pump for 2 h. (2) Temperature and stress setting and seepage test: The triaxial pressure chamber was placed in a 30 °C constant-temperature water bath. Axial pressure and confining pressure were alternately loaded to predetermined values (axial pressure: 12 MPa; confining pressure: 3 MPa) at a gradient of 0.5 MPa by force. The inlet valve was opened to inject CH4 until the inlet gas pressure reached to 0.3 MPa. In the state, the sample was allowed to adsorb gas until adsorption equilibrium. The gas pressure valve was adjusted to raise pressure in the following order: 0.3 MPa→0.55 MPa→0.8 MPa→1.05 MPa→ 1.3 MPa→1.55 MPa→1.8 MPa→2.05 MPa→2.3 MPa. The deformation and flow rate values for adsorption equilibrium and flow rate stability were recorded under each gas pressure. Then, both the gas pressure and the confining pressure were raised to ensure constant effective stress of 5.7 MPa. (3) Replacement of specimen: The previous 3 steps were repeated at 40 °C, 50 °C and 60 °C, respectively. (4) Parallel test: The parallel test of He was carried out with reference to the previous 3 steps.

2.2. Experimental apparatus The device used in the isothermal adsorption experiment is the HCA high-pressure volumetric method adsorption system that can be used to carry out isothermal adsorption test under different conditions of temperature and pressure (Li and Kang, 2016). As exhibited in Fig. 1, the device used in the seepage experiment is the triaxial servo-controlled seepage equipment for thermo-fluid-solid coupling of methanecontaining coal which can be used to carry out mechanical and permeability tests under different conditions of temperature, stress and gas pressure (Yin et al., 2013). 2.3. Experimental procedure 2.3.1. Isothermal adsorption measurement The adsorption characteristics of coal are affected by temperature and gas pressure. In order to investigate the effects of temperature and excess adsorption on the adsorption characteristics of coal, the isothermal adsorption experiments at 30 °C, 40 °C, 50 °C and 60 °C were carried out by selecting CH4 as the test gas in this experiment and setting 7 adsorption equilibrium points in the range of 0.1 MPa–4.7 MPa. In this way, the adsorption capacity under different pressure points was determined. Specific test steps are as follows: Table 1 Proximate analysis of coal. Number

Moisture/%

Ash/%

Volatile matter/%

Fixed carbon/%

1 2 3 Average

1.51 1.59 1.48 1.53

38.14 37.70 35.68 37.20

11.43 11.06 11.58 11.36

48.92 49.56 51.26 49.91

3

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Table 2 Distribution of pores and fractures in coal. Number

Porosity/%

Maximum pore and fracture radius/ μm

Minimum pore and fracture radius/ μm

average pore and fracture radius/ μm

Total number of pores and fractures

1 2 3 Average

2.31 2.74 2.69 2.58

247.96 194.62 180.36 207.65

11.65 11.98 13.64 12.42

44.95 46.78 57.04 49.59

268 336 301 302

16n

m

16P

ρg = V = V = RT (Weniger et al., 2010) where T is the temperature, K; R is the universal gas constant; V is the volume of bulk gas phase, cm3; P is the gas pressure, MPa; n is the amount of substance, mol; and ρabs, which can take the value of 0.423 g/cm3, is the density of adsorption phase (Weniger et al., 2010). The Langmuir equation, which is the most commonly used adsorption model (Langmuir, 1916), can be expressed as: q=

abp 1 + bp

(2)

where q is the adsorption amount under the equilibrium pressure (P), cm3/g; a and b are the adsorption constants. Based on the Langmuir equation, the absolute adsorption amount considering the effect of temperature can be expressed as (Tang et al., 2016):

qabs = a⋅

(3)

where K(T) is the equilibrium constant that changes with temperature. Since gas adsorption is an exothermic reaction, the heat produced is normally dissipated toward the immediate surroundings. The ratio of the change in the adsorbate enthalpy to the change in the adsorbed amount is normally termed the heat of adsorption. The adsorption equilibrium constant, K(T), normally represented by the Van't Hoff expression, is temperature-dependent (Ji et al., 2014; Fianu et al., 2018):

Fig. 1. Schematic diagram of triaxial servo-controlled seepage equipment for thermo-fluid-solid coupling of methane-containing coal.

E K (T ) = A0 ⋅exp⎛− a ⎞ ⎝ RT ⎠

3. Adsorption-permeability model of coal under stress and temperature coupling condition

(4) 3

where A0 is the pre-exponential coefficient, cm /g; and Ea is the adsorption energy, kJ/mol. By substituting Eq. (3) into Eq. (1), Eq. (1) can be rewritten as follows:

3.1. Langmuir model considering the correction of temperature and excess adsorption Excess adsorption capacity refers to the adsorption amount beyond the gas phase density in the adsorption phase (Weniger et al., 2010). The absolute adsorption capacity, which is calculated directly from results of the equivalent adsorption test, is the actual gas adsorption capacity (Tian et al., 2016). The volumetric method is usually used for the isotherm adsorption test of coal in the laboratory. The principle of volumetric method is to calculate the difference in methane amounts before and after adsorption by obtaining the amounts through the equation of state of an ideal gas (PV = nRT). However, this process assumes the free space volume (Vf) to be constant before and after adsorption and ignores the pore volume occupied by methane adsorption phase (Va) (see Fig. 2). As the adsorption phase volume cannot be experimentally measured, the adsorption amount measured by the experiment is actually the excess adsorption capacity (Li et al., 2018). The relation between excess adsorption capacity and absolute adsorption amount can be obtained by Eq. (1) (Tang et al., 2016):

ρg ⎞ qex = qabs ⎜⎛1 − ⎟ ρabs ⎠ ⎝

K (T )⋅P 1 + K (T )⋅P

qex = a⋅

ρg ⎞ K (T )⋅P ⎛ ⋅⎜1 − ⎟ 1 + K (T )⋅P ⎝ ρabs ⎠

(5)

3.2. Deformation induced by temperature 3.2.1. Adsorption deformation induced by temperature The surface energy of coal matrix decreases with the adsorption of methane. Matrix swelling is directly proportional to the decrease of surface energy. According to the Gibbs formula, the reduction of surface free energy caused by adsorption, π, is (Adamson, 1990):

∫P

π = RT

P

Γd (ln P )

(6)

0

where Γ is the surface excess, mol/m , namely, the difference between the gas concentration of the surface of coal and that of the interior of coal. 2

Γ is mathematically defined as (Adamson, 1990):

(1)

Γ=

where qex is the excess adsorption quantity, cm3/g; qabs is the absolute adsorption quantity, cm3/g; ρg is the density of bulk gas phase, g/cm3. According to equation of gas state PV = nRT, ρg can be expressed as

qex Vm S

(7) 2

where S is the specific surface area of coal, m /g; and Vm is the standard 4

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Fig. 2. Schematic diagram of test error caused by the volumetric method.

distribution characteristics of coal (Jin et al., 2017). The distribution of the size of pore fissure can be expressed by the fractal theory (Zhu et al., 2011a):

molar volume, L/mol. The solid strain is proportional to the adsorption-induced reduction of surface free energy (Bangham and Fakhoury, 1931). The linear strain can be expressed as:

ΔL = γ ⋅π L

Nl = Nl0 (l/ l max )−Df

where Nl0 is the number of micro-cracks of per unit volume; Nl is the cumulative number of micro-cracks whose feature sizes are equal to or greater than l; l is the feature size of micro-cracks, m; lmax is the largest feature size; and Df is the fractal dimension of pores in coal. In the twodimensional space, 0＜Df＜2. In the three-dimensional space, 0＜Df＜3. The fractal dimension is linearly related to the variation of temperature, which can be expressed as (Nakagawa et al., 2000):

(8)

where L is the width of coal, m; and r is the deformation constant. The relationship between the deformation constant and the mechanical properties of coal can be expressed as (Bangham and Fakhoury, 1931):

γ=

Sρc (9)

EA

Df = Df 0 + λ (ΔT / T0)

where ρc is the density of coal, g/cm ; and EA is the modulus of adsorption-induced expansion of coal, MPa. By substituting Eqs. (6), (7) and (9) into Eq. (8), the value of adsorption-induced strain under temperature is obtained as (Liu and Harpalani, 2013): 3

εs =

3ρ RT 3ΔL = c L Vm EA

∫P

P

0

(10)

εs =

3ρc aRT Vm EA

1 + K (T ) P

ln1 + K (T ) P0 −

⎛ P − P0 − Vm EA Rρabs ⎜⎜ ⎝ 48aρc

K (T )

⎞ ⎟⎟ ⎠

εTp =

3 − Df

φm0

(15)

ΔVmp Vm

= φm − φm0 =

φm0 − φm0 1 − ϑΔT

(16)

where εTp is the volumetric strain produced by the thermal cracking; and ϑ can be expressed as Cf = γD . The deformation of coal matrix is jointly influenced by the above three aspects under the condition of constant effective stress and different temperatures. Temperature could affect the adsorption of coal, the increase in pore fissures and the thermal expansion of matrix. Hence, the volumetric strain of coal can be given as follows:

(11)

3.2.2. Expansion deformation caused by temperature The coal undergoes expansion deformation under the influence of temperature, which affects the internal structure of coal. The deformation caused by thermal expansion can be expressed as (Wu et al., 2011):

εe = αT ΔT

3 − Df 0

where φm0 is the initial porosity. Combining Eq. (14) and Eq. (15), Eq. (16) can be obtained:

where εs is the adsorption-induced volumetric strain; and p0 is the initial gas pressure, MPa. Through integration, Eq. (10) can be rewritten as follows: 1 + K (T ) P ln1 + K (T ) P0

(14)

where Df0 is the initial fractal dimension; λ is the fractal sensitivity coefficient; and T0 is the room temperature, available at 298 K. When the fractal dimension is small, the porosity of coal matrix at different pore pressures can be expressed as (Teng et al., 2016a):

φm =

qex dP p

(13)

εm = εs + εTp + εe

(17)

(12)

where εe is the volumetric strain produced by the thermal expansion; αT is the coefficient of thermal expansion, K−1; and ΔT is the variation of temperature, K.

εm =

3ρc aRT Vm EA

1 + K (T ) P

ln1 + K (T ) P0 −

1 + K (T ) P

⎛ φm0 ln1 + K (T ) P0 ⎞ P − P0 − + ⎜ ⎜ Vm EARρabs K (T ) ⎟⎟ 1 − ϑΔT ⎝ ⎠ 48aρc

− φm0 + αT ΔT

3.2.3. Thermal cracking Thermal cracking refers to the appearance of a series of defects and micro cracks in the interior of coal caused by the change in temperature. Coal, a dual-porosity medium, has a complex fractal pore structure. Fractal behavior determines the fractal dimension (Jin et al., 2019) which, however, differs a little due to difference in the experimental value. Meanwhile, the pore size distribution, pore structure and tortuosity of coal affect its permeability (Cai et al., 2018), and the fractal dimension can only approximately characterize the pore size

(18)

3.3. The variation of permeability caused by the slippage effect and adsorption swelling When the effective stress is constant, the measured or apparent permeability, kg, can be expressed as (Zou et al., 2016):

kg = k 0 + Δk s + Δkb where k0 is the absolute permeability, 10 5

(19) −3

μm ; Δkb is the variation of 2

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permeability caused by the slippage effect, 10−3 μm2; Δks is the variation of permeability caused by adsorption swelling, 10−3 μm2. The measured permeability considering slippage effect can be expressed as (Klinkenberg, 1941):

kg = k 0 (1 +

B ) P

(20)

where B is the klinkenberg coefficient. The klinkenberg coefficient of He, BHe, can be obtained by Eq. (20), while that of CH4 can be expressed as (Wang et al., 2014):

BCH4 =

w He μCH4 wCH4 μ He

MHe BHe MCH4

(21)

where μ is the kinetic viscosity of the fluid, Pa•s; M is the fluid molecular weight; and w is the channel width of coal, m. Under the pore pressure of p, the fracture width of coal, wp, is expressed as:

wp = w0 + win

(22)

where w0 is the initial fracture width of coal, m; and win is the variation p of fracture width, win = ∫p Δwm ∂p. 0 where Δwin can be given as (Cui and Bustin, 2005):

Δwm = −dfh 0 dεm

(23)

where f is a modified parameter for coal deformation; and h0 is the initial width of the coal matrix, m. Combining Eq. (22) and Eq. (23), Eq. (24) can be obtained:

wCH4 = w0 − h 0 f

⎧ 3ρ aRT 1 + K (T ) P ⎪ Vc E ln1 + K (T ) P0 − m A

⎨ ⎪+ ⎩

φm0 1−ϑΔT

− φm0

⎛ P − P0 − Vm EARρabs ⎜ ⎝ + αT ΔT 48aρc

1 + K (T ) P

ln1 + K (T ) P0 K (T )

Fig. 3. Dual-porosity model of idealized coal (Wang et al., 2014).

⎞⎫ ⎟⎪ ⎠⎬ ⎪ ⎭

effective stress and swelling deformation under the boundary condition of triaxial stress (Lu et al., 2016):

k E = exp ⎧−3Cf ⎡ (σ − σ0) − (P − P0) + f εs⎤ ⎫ ⎢ ⎥⎬ ⎨ k0 3(1 − 2 υ ) ⎣ ⎦ ⎩ ⎭

(24) Based on the cubic model, Eq. (25) can be obtained:

φ=

3w h

where Cf is the compression coefficient of fracture, MPa ; k0 is the initial permeability of coal, 10−3 μm2; E is the elastic modulus, MPa; and υ is Poisson's ratio. When the effective stress is constant, the increment of effective stress in three directions all equals to zero, (σ − σ0) − (P − P0) = 0 . Through integration, Eq. (29) can be rewritten as follows:

(25)

By solving Eqs. (21), (24) and (25), Eq. (26) can be acquired:

μCH4

1

BCH4 = 1−

3 f φ0

⎧ 3ρ aRT 1 + K (T ) P ⎪ Vc E ln1 + K (T ) P0 − m A

⎨ ⎪+ ⎩

φm0 1−ϑΔT

− φm0

⎛ P − P0 − Vm EARρabs ⎜ ⎝ + αT ΔT 48aρc

MHe BHe MCH4

1 + K (T ) P ln1 + K (T ) P0

K (T )

⎞⎫ ⎟⎪ ⎠⎬ ⎪ ⎭

μ He

⎧ ⎡ 3ρ aRT 1 + K (T ) p 48aρc ⎛ k E ⎢ c p − p0 ln1 + K (T ) p0 − = exp −3Cf f ⎨ k0 3(1 − 2υ) ⎢ V0 EA V0 EA Rρads ⎜⎜ ⎝ ⎣ ⎩ 1 + K (T ) p

The variation of permeability caused by the slippage effect can be expressed as:

B BCH4 ⎞ − k 0 = k 0 CH4 P P ⎠

BCH4 P

⎞⎤ ⎫ ⎟⎥ ⎬ ⎟⎥ ⎠⎦ ⎭

(30)

Slippage effect is an important factor affecting the coal permeability. Considering the influence of slippage effect, Eq. (31) can be obtained by substituting Eq. (30) into Eq. (20):

(27)

⎧ ⎡ 3ρc aRT 1 + K (T ) P 48aρc ⎛ k E ln1 + K (T ) P0 − P − P0 = exp −3Cf f ⎢ ⎨ k 3(1 − 2υ) ⎢ Vm EA Vm EARρabs ⎜⎜ ⎝ ⎣ ⎩

By substituting Eq. (27) into Eq. (19), Δks, the variation of permeability caused by adsorption swelling can be expressed as:

Δks = k g − k 0 − k 0

ln1 + K (T ) p0 K (T )

−

(26)

Δkb = kCH4 ,g − k 0 = k 0 ⎛1 + ⎝

(29) −1

1 + K (T ) P

(28)

−

ln1 + K (T ) P0 K (T )

3.4. Improved permeability model

⎞⎤ ⎫ B ⎟⎟⎥ ⎬ ⎛1 + P ⎞ ⎠ ⎥ ⎝ ⎠⎦ ⎭

(31)

Xue et al. (2016) combined the compressibility coefficient of fracture with the damage of coal:

Mechanically, coal is in the elastic state during the process of deep CBM production. Coal is a porous fractured medium composed of coal matrixes and fractures. The bundled matchstick conceptual model has been widely used as a basis to describe the coal cleat system and to derive a number of permeability models (Pan and Connell, 2012), as shown in Fig. 3. Eq. (29) is the permeability equation considering the

Cf = γD

(32)

where γ is defined as the mutation coefficient, which reflects the sensitivity of fracture development of coal to effective stress; D is the damage variable. 6

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Coal is affected by pore pressure and temperature, so its elastic modulus will change in the mechanical process (Xu et al., 2018). The elastic modulus is generally used to characterize coal damage under temperature effect. Since the study aims to investigate the effect of temperature on coal permeability, the temperature-induced elastic modulus variation can be used to characterize coal damage under the action of temperature. Because the elastic modulus of coal was tested at 25 °C, 30 °C, 40 °C, 50 °C and 60 °C, this study defined the thermal damage of coal at room temperature (25 °C) to be 0 and regarded it as the reference temperature of damage variable (without considering the initial damage of coal). With the damage variable reflected by elastic moduli, and the thermal damage of coal is defined as (Xu et al., 2018):

DT = 1 −

ET E0

(33)

where ET and E0 de note elastic moduli at the temperature T and 25 °C, respectively, MPa; and DT is the damage variable at T. Substituting Eqs. (32) and (33) into Eq. (31), the permeability model under stress and temperature coupling condition can be obtained:

⎧ ⎡ 3ρc aRT 1 + K (T ) P 48aρc ⎛ E k = k 0 exp fγT DT ln1 + K (T ) P0 − P − P0 ⎢ ⎨ (1 − 2υ) ⎢ Vm EA Vm EARρabs ⎜⎜ ⎝ ⎣ ⎩ 1 + K (T ) P

−

ln1 + K (T ) P0 K (T )

⎞⎤ ⎫ BCH4 ⎟⎟⎥ ⎬ ⎛1 + P ⎞ ⎠ ⎥ ⎝ ⎠⎦ ⎭

(34)

where γT is the temperature mutation coefficient which reflects the sensitivity of coal fracture development to temperature under constant effective stress. Based on the permeability model established by Lu et al. (2016), this study established a permeability model considering the effects of temperature under effective stress and slippage by introducing the concept of damage variable. 4. Results and discussion 4.1. Analysis of adsorption characteristics of coal Fig. 4. Isothermal adsorption curves at different temperatures of (a) Songzao coal and (b) Datong anthracite (Xing et al., 2015).

Applicability of the improved model and the Langmuir model in different gas pressure stages at various temperatures was verified by experimental data and citing the isothermal adsorption test results of Xing et al. (2015). The adsorption quantities of coal at different temperatures are shown in Fig. 4. The modified Langmuir model (Eq. (5)) is used to calculate the adsorption capacity and is compared with the Langmuir model (Eq. (2)). The parameters are listed in Table 3. (1) As can be seen in Fig. 4, the adsorption capacity of coal grows with the rise of pore pressure at the low-pressure stage. As the pore pressure increases to 3 MPa, the adsorption rate gradually decreases, and the coal gradually reaches adsorption saturation. When the pore pressure is raised to 4.7 MPa, the adsorption amounts at 30 °C, 40 °C, 50 °C and 60 °C are 26.62 cm3/g, 26.41 cm3/g, 25.28 cm3/g, 23.35 cm3/ g, respectively. With the rise of gas pressure, the excess adsorption capacity increases at the beginning of high-pressure stage, whereas it decreases after the gas pressure exceeds 8 MPa. The reason is that the adsorption capacity measured by volumetric method often ignores the volume occupied by the adsorption phase. Besides, at each pressure point, the adsorption capacity at a lower temperature is greater than that at a higher temperature. The reason is that the molecular activity of gas grows stronger at a higher temperature, so that the adsorption of coal becomes more difficult and the adsorption quantity is reduced gradually (Zhang, 2008). (2) In Table 3, The AAD for Langmuir model was higher in most of the results compared with the modified Langmuir model. This shows that the relative error is small for the modified Langmuir model. The p-

value for Langmuir model was a little higher in most of the results compared with the modified model. It explained that the significant difference is small for the modified Langmuir model, and showed the advantage of the modified model in prediction. When the temperature is 30 and 40 °C in Songzao coal, the correlation coefficient (R2) of the Langmuir model is equal to the Modified Langmuir model. As the temperature increases, the R2 of modified model is higher than the Langmuir model. At the low-pressure stage, the two models, which correspond well to the test results, can both reflect the relationship between adsorption quantity and gas pressure at different temperatures. However, at the high-pressure stage, the modified Langmuir model considering temperature and excess adsorption capacity boasts higher fitting accuracy. To be specific, under low gas pressure, the density of gas free phase is small, and the error caused by the volume of adsorption phase can be ignored (Li et al., 2018). The Langmuir model can also well characterize adsorption characteristics of coal. However, gas in free phase gradually becomes denser with the continuous rise of gas pressure. Accordingly, the excess adsorption capacity expands gradually due to the error. In this case, the Langmuir model will cause a larger error, while the improved adsorption model better fits the data in both low-pressure and high-pressure stages. In addition, it is reasonable to consider the effect of temperature which is ignored by the Langmuir model. With the rise of temperature, the maximum adsorption 7

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Table 3 Comparison of model parameters. Coal sample

T/°C

Langmuir model 3

Songzao coal

Datong anthracite

30 40 50 60 40 60 80

1

a (cm ·g- )

b (MPa

30.58 30.58 30.12 28.99 18.23 16.65 16.15

1.27 1.11 0.97 0.77 3.87 2.18 1.71

Modified Langmuir model −1

)

R

2

0.996 0.994 0.994 0.992 0.809 0.924 0.962

p-value

AAD/%

a (cm3·g−1)

A0 (cm3·g-1)

Ea (kJ·mol−1)

R2

p-value

AAD/%

3.47E-07 1.17E-06 6.97E-07 6.68E-07 7.35E-05 1.05E-06 5.12E-08

4.14 5.12 4.83 4.93 5.42 4.84 3.97

34.06 34.41 34.13 33.80 21.18 20.79 17.58

0.0006 0.0011 0.0079 0.1020 0.0021 0.0015 0.0037

18.47 17.11 12.09 4.64 17.66 17.24 16.88

0.996 0.994 0.996 0.997 0.943 0.973 0.980

6.88E-08 2.50E-07 9.37E-08 4.39E-08 6.50E-07 3.25E-10 1.80E-09

4.13 4.95 4.73 4.69 4.02 3.68 3.66

temperatures, macroscopically reflect the deformation laws of gas-adsorbing coal under the coupling of stress and temperature.

quantities, namely the values of a, calculated by the two models both decrease, which is consistent with the previous study (Guo et al., 2015). The intrinsic energy of gas molecules increases with the rise of temperature and overcomes the physical adsorption. As a result, the adsorbed gas gradually desorbs, and the value of a gradually declines. Furthermore, the comparison between the two models indicates that the modified Langmuir model has a larger value of a, because it considers the influence of adsorbed phase volume. In contrast, the adsorption capacity measured in the laboratory is lower because it neglects the influence (Weniger et al., 2010). Therefore, the relatively large value calculated by the modified model is the real maximum adsorption capacity of coal.

4.3. Relationship between deformation, permeability and pore pressure of coal The gas adsorption of coal leads to swelling deformation, which causes changes in fractures and pores within the coal and affects the permeability of coal (Li et al., 2017). Fig. 6 shows the measured relationship among flow rate (Q), strain and pore pressure. Fig. 7 shows the relationship between permeability and pore pressure at different temperatures. (1) Under the condition of constant effective stress, the radial strain decreases with the increase of pore pressure at all temperatures. Besides, the axial strain decreases at lower temperatures (30 °C and 40 °C) and increases at higher temperatures (50 °C and 60 °C). The coal adsorbs gas to produce expansion stress, so both the axial strain and the radial strain are gradually lowered. With the rise of temperature, the strength of coal falls gradually, and the inhibitory effect of temperature on the adsorption of coal matrix is gradually enhanced (Guan et al., 2018). Moreover, the increase of desorption amount leads to the shrinkage and deformation of coal, which results in the gradual increase of axial strain. The expansion stress is much lower than the axial pressure but approximates the confining pressure, so the radial strain keeps decreasing. (2) During gas loading process, the flow rate of gas increases gradually while the volumetric strain decreases gradually on the opposite. When the pore pressure rises from 0.3 MPa to 2.3 MPa, the flow rate increases by 1.345 cm2/s, 1.169 cm2/s, 1.581 cm2/s and 1.052 cm2/ s at temperatures of 30 °C, 40 °C, 50 °C and 60 °C, respectively. The increase rate of flow rate grows with the rise of pore pressure. The reason is as follows: With the rise of pore pressure, the gradual increase in the pressure difference between both ends of the sample promotes the flow of gas. When the coal matrix adsorbs gas, volumetric strain decreases gradually. At the initial stage of loading, the effective seepage channels of coal shrink and the flow rate increase slowly due to the adsorption. With the continuous rise of pore pressure, the adsorption of coal matrix is saturated, and meanwhile the increase rate of gas flow rate goes up gradually. (3) Under constant effective stress, the permeability gradually goes down with the rise of pore pressure, and the permeability curves goes up with the rise of temperature. The reason is as follows: The slippage effect is strong under low pore pressure; as pore pressure rises, it gradually weakens, and the coal adsorbs gas to swells. As a result, the pores close gradually, which results in gradual reductions in effective seepage channel and permeability. During gas loading, the slippage effect gradually disappears and the adsorbed gas saturates coal. The increase of pore pressure has a very limited effect on pore expansion, so the permeability tends to level off at the later stage of pore pressure rise. The permeability of coal increases as a whole with the rise of temperature. The adsorption swelling,

4.2. Calculated value of adsorption deformation To study the law of adsorption deformation of coal during gas pressure variation at different temperatures, the model of coal deformation with the consideration of temperature and excess adsorption was established. The strain under stress and temperature can be obtained by using Eq. (11). Fig. 5 shows the relationship between the adsorption deformation and pore pressure at different temperatures. The adsorption-induced strain decreases gradually with the rise of pore pressure. At the pore pressure of 2.3 MPa, the adsorption-induced strain decreases to 0.0087, 0.0084, 0.0080 and 0.0070 at temperatures of 30 °C, 40 °C, 50 °C and 60 °C, respectively. The adsorption-induced strain decreases gradually with the rise of temperature due to the inhibitory effect of temperature on coal adsorption. With the rise of temperature, the adsorption capacity of coal gradually decreases, resulting in the gradual reduction in both the swelling effect and the strain of coal. The calculated results, which can indicate the relationship between coal deformation and pore pressure at different

Fig. 5. Relationship between adsorption deformation and pore pressure at different temperatures. 8

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Fig. 6. Relationship among flow rate, strain and pore pressure at (a) 30 °C, (b) 40 °C, (c) 50 °C and (d) 60 °C.

of various factors. 4.4. Influence of adsorption swelling and slippage effect on permeability The permeability of coal is mainly affected by the coupling effect of adsorption swelling and slippage effect under the condition of constant effective stress. The slippage factor was calculated by substituting parameters listed in Table 4 into Eq. (26). Furthermore, the permeability variations caused by the adsorption swelling and slippage effect at different temperatures were obtained according to Eqs. (27) and (28). The coal porosity and adsorption constants in Table 4 were measured by the authors in person, while the parameters of thermal expansion and fractal were obtained from literatures with corresponding conditions of temperature and sample, so as to ensure the rationality. EA, the expansion modulus of coal, does not exactly equal the elastic modulus. Through mechanical tests of several different kinds Table 4 Reference parameters of the equations.

Fig. 7. The relationship between permeability and pore pressure at different temperatures.

thermal expansion, thermal cracking and slippage effect of coal caused by the rise of temperature comprehensively affect the permeability change (Teng et al., 2016b). The desorption ability of coal and slippage effect gradually increase with the rise of temperature (Pan et al., 2012). At the same time, the heating of coal will generate new pores and fractures (Balankin and Espinoza, 2012), further promoting the permeability of coal. However, when the temperature rises to 60 °C, the permeability of coal ceases increasing obviously. The reason is that the rise of temperature aggravates the thermal expansion of coal (Wang et al., 2014). The internal expansion of coal occurs under the action of external force, and the crack of coal pores is gradually closed. Resultantly, coal permeability increases at a lower rate under the comprehensive influence 9

Parameter

Value

Source

Φm0 PL, MPa c1, MPa−1 c2, K−1 EA, MPa T0, K aT, K−1 λ Df0 f Φ0 M CH4 M He ρc, g/cm3 ϑ

0.05 1.57 0.07 0.02 1900 298 2.4 × 10−5 0.46 2.83 0.255 0.04 16 4 1.6 0.009

Teng et al. (2016b) Wu et al. (2011) Zhu et al. (2011b) Zhu et al. (2011b) Liu et al. (2013) Assumed Zhu et al. (2011b) Nakagawa et al. (2000) Teng et al. (2016b) Chen et al. (2012) Measured

Measured Calculated

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the pore pressure is loaded slowly from 0.3 MPa to 2.3 MPa, the permeability variations caused by the slippage effect at temperatures of 30, 40, 50 and 60 °C decrease by 0.008 × 10−3, 0.016 × 10−3, 0.027 × 10−3 and 0.039 × 10−3 μm2 at reduction rates of 84.86%, 85.01%, 85.12%, 85.31%, respectively, because the rise of pore pressure gradually reduces the mean free path of molecules and thus weakens the slippage effect (Zou et al., 2016). In addition, the permeability variation caused by the slippage effect increases with the rise of temperature, because the gas density decreases with temperature, leading to the expansion of gas volume and the increase of mean free path. As a result, both the slippage effect and the permeability variation increase (Qu et al., 2012). (2) The permeability variation caused by the adsorption swelling of coal first drops sharply and then levels off with the increase of pore pressure. When the pore pressure rises from 0.3 MPa to 2.3 MPa, the permeability variations caused by the adsorption swelling at temperatures of 30, 40, 50 and 60 °C decrease by 0.072 × 10- 3, 0.142 × 10- 3, 0.257 × 10- 3 and 0.28 × 10- 3 μm2, respectively. When it rises from 0.3 MPa to 0.55 MPa, the variations account for 75.13%, 48.29%, 61.16% and 61.78% of the total variation, respectively. With the increase of pore pressure, the gas adsorption on coal leads to the expansion of coal matrix, thus gradually reduces the effective seepage channel as well as the permeability. When the pore pressure rises to 0.55 MPa, the adsorption capacity of coal is strong, and the permeability reduction caused by adsorption swelling falls sharply. As the pore pressure continues to increase, the adsorption capacity of coal and the permeability keep dropping, which is consistent with the previous findings (Wang et al., 2010). In addition, the permeability variation increases with the rise of temperature. The adsorption-induced swelling effect is gradually weakened, while the shrinkage effect of coal matrix becomes more obvious. Therefore, this paper theoretically proves that both the coal permeability and the relative variation of permeability gradually goes up with the rise of temperature.

Table 5 Slippage factor value of He at different temperatures. T/°C

30

40

50

60

BHe

0.135

0.303

0.370

0.780

Fig. 8. The relationship between the pore pressure and the permeability variation caused by the slippage effect.

4.5. Comparison of permeability models The thermal damage (DT) of coal was calculated by substituting the elastic modulus of different temperatures (30, 40, 50 and 60 °C) in Table 6 into Eq. (32). The model curves of permeability and pore pressure were calculated by substituting bCH4 and DT into Eq. (34) permeability model, and correctness of the model was verified through a comparison with the test results. To further prove applicability of the modified model established in this study, it was compared with S&D and C&B models. The parameters of model fitting are given in Table 7. Fig. 10 displays curves of model comparison.

Fig. 9. The relationship between the pore pressure and the permeability variation caused by the adsorption swelling.

(1) Compared with the S&D and C&B models, the curves calculated by the modified model are in better agreement with the experimental results and can well reflect the relationship between pore pressure and permeability at different temperatures. In fact, the S&D and C& B models cannot well represent the evolution trend of coal permeability at different temperatures, as they are suitable for the uniaxial strain condition (Shi and Durucan, 2004) without considering the effect of temperature on coal permeability. Under constant effective stress, the permeability of coal drops rapidly at first and then declines slowly with the rise of pore pressure. However, the permeability curves calculated by S&D model and C&B model show a trend of decreasing first, then rising and finally

of coal, Maggs (1946) pointed out that the elastic modulus and expansion modulus of coal were not equal, and that the expansion moduli of different kinds of coal did not differ significantly. This paper referred to the expansion modulus (1900 MPa) of two kinds of coal samples studied by Levine (1996) and Moffat and Weale (1955). a and K(T) were measured by an isothermal adsorption test; corresponding values of μCH4 and μHe were taken according to different temperature and pressure conditions; ρc was determined to be 1.6 g/cm3 by a true density test; and BHe was obtained by substituting test data into Eq. (20) (see Table 5). Fig. 8 presents the relationship between the pore pressure and the permeability variation caused by the slippage effect. Fig. 9 exhibits the relationship between the pore pressure and the permeability variation caused by the adsorption swelling.

Table 6 Values of elastic moduli under different temperature conditions.

(1) In the gas loading process, the permeability variation caused by slippage effect gradually decreases. When the pore pressure exceeds 1.05 MPa, the permeability variation becomes gradually steady. As 10

T/°C

25

30

40

50

60

E/MPa

181.51

178.88

173.74

168.46

164.11

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adsorption capacity increases at the beginning of high-pressure stage, whereas it decreases after the gas pressure exceeds 8 MPa. Besides, it decreases gradually with the rise of temperature. At the high-pressure stage, the modified Langmuir model considering temperature and excess adsorption capacity can better characterize the adsorption characteristics of coal. (2) With the rise of pore pressure, the gas flow rate increases gradually and the volumetric strain and radial strain decrease. In addition, the axial strain at low and high temperatures shows the opposite upward and downward trends. When the effective stress is constant, the permeability gradually decreases with the increase of pore pressure, and the permeability curves goes up with the rise of temperature. The adsorption-induced strain gradually increases with the rise of pore pressure and decreases with the rise of temperature. (3) The slippage effect gradually diminishes with the increase of pore pressure, and the permeability variation caused by the slippage effect increases with the rise of temperature. During gas loading process, the permeability variation caused by the adsorption swelling of coal drops sharply and then tends to level off. When the pore pressure rises to 0.55 MPa, the adsorption capacity of coal is gradually reduced. Besides, the permeability variation gradually increases with the rise of temperature, and it has a greater effect when the pore pressure is lower. (4) A model of coal permeability considering the slippage effect under stress and temperature coupling condition was established. Compared with the S&D and C&B models, the curves calculated by the modified model are in better agreement with the experimental

Table 7 Comparison of model parameters. T/°C

30 40 50 60

Modified model

S&D model

C&B model

γT /MPa−1

AAD /%

Cf /MPa−1

AAD /%

Cf /MPa−1

AAD /%

34.15 18.45 11.55 11.15

4.31 6.54 8.01 9.07

0.76 1.47 1.58 0.98

11.15 24.42 53.41 131.32

−0.16 −0.35 −0.37 −0.42

13.36 21.83 22.59 23.55

decreasing. Therefore, given their poor applicability of theoretical mechanism and fitting with experimental results, neither the S&D model nor the C&B model can be applied to the effective stress condition. (2) At different temperatures (30 °C, 40 °C, 50 °C and 60 °C), the temperature mutation coefficients calculated by the modified model are 34.15 MPa−1, 18.45 MPa−1, 11.55 MPa−1 and 11.15 MPa−1, respectively. γT decreases with the increase of temperature. This indicates that the sensitivity of fracture development to temperature decreases, and the effect of temperature on coal permeability weakens, which can also be reflected by the permeability curve at different temperatures. 5. Conclusions (1) The adsorption capacity of coal grows with the rise of pore pressure at the low-pressure stage. With the rise of gas pressure, the excess

Fig. 10. Curves of permeability model comparison at (a) 30 °C, (b) 40 °C, (c) 50 °C and (d) 60 °C. 11

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results and can well simulate and explain the law of seepage evolution with the increase of mining depth. The sensitivity of fracture development to temperature decreases, and the effect of temperature on coal permeability weakens with the rise of temperature.

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